Bryan Pearsaul
Jesse Phillips
Discrete Structures II
Reducibility
To reduce a problem A to a problem B is to show that if you had a solution to B then you
can also solve A.
Given: A
TM
= {<M, w>  M is a TM and it accepts w}
A
TM
is undecidable.
Prove:
HALT
TM
= {<M, w>  M is a TM and halts on w} is also undecidable.
Goal: If we can reduce A
TM
to HALT
TM
, then HALT
TM
is undecidable as well.
1.
Assume a TM exists that decides HALT
TM
, call this R.
2.
Produce S: a machine that decides A
TM
S has input (M, w)
1). Call R(M, w)
2). If R rejects,
reject (because we know that w
∈
L(M))
else
run M on w and return its answer
Therefore we have shown A
TM
to be decidable, which we know is not true. So our initial
assumption is false.
Prove:
E
TM
= {<M>  M is a TM and L(M) =
Ø
) is undecidable.
Goal: If we can reduce A
TM
to E
TM
, then E
TM
is undecidable as well.
1.
Assume a TM exists that decides E
TM
, call this R.
2.
Produce a machine to solve A
TM
(We’re stuck if we just pass <M> to R, so we must pass a different machine as a
parameter)
// we’re never going to run M
’
, we will just use it as input to R.
Create a new machine M
’
from M and w whose language is L(M
’
) = {
Ø
, {w}}.
1). M
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 Spring '08
 Staff
 Halting problem

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